Representation of cyclotomic fields and their subfields

Let $mathbb{K}$ be a finite extension of a characteristic zero field $mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $mathbb{F}$ represents $mathbb{K}$ if $mathbb{K} cong {{mathbb{F}left[ A right]} mathord{left/ {vphantom {{mathbb{F}left[ A right]} {leftlangle B rightrangle }}} right. kern-0em} {leftlangle B rightrangle }}$ , where $mathbb{F}left[ A right]$ denotes the subalgebra of $mathbb{M}_n left( mathbb{F} right)$ containing A and 〈B〉 is an ideal in $mathbb{F}left[ A right]$ , generated by B. In particular, A is said to represent the field $mathbb{K}$ if there exists an irreducible polynomial $qleft( x right) in mathbb{F}left[ x right]$ which divides the minimal polynomial of A and $mathbb{K} cong {{mathbb{F}left[ A right]} mathord{left/ {vphantom {{mathbb{F}left[ A right]} {leftlangle {qleft( A right)} rightrangle }}} right. kern-0em} {leftlangle {qleft( A right)} rightrangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.